Jacobson radicals, abelian $p$-groups and the ${\oplus }_c$-topology

Abstract

In studying the Jacobson radical of the endomorphism ring of a separable abelian $p$-group, Sands (1984) identified the useful Condition (C). Our central result is that the group $G$ satisfies Condition (C) precisely when it is complete in its ${\oplus }_c$-topology, which uses the set of subgroups $X\le G$ such that $G/X$ is a direct sum of cyclics as a neighborhood base of 0. This equivalence is then used to compute the Jacobson radicals of the endomorphism rings of a variety of such groups, including those in the so-called Keef class. It is shown that Sands’ attempt to “complete” an arbitrary group with respect to Condition (C) is equivalent to D’Este’s (1980) attempt to show that the ${\oplus }_c$-topology is completable. Since Mader (1983) provided a counterexample to D’Este’s result, it follows that Sands’ result also fails.